Polarisation effects in the theory of optical solitons

Abstract

This thesis examines the properties of light beams that are able to trap themselves and propagate without diffraction in nonlinear Kerr materials. Beams of this type, called spatial solitary waves, have potential applications to all-optical switching devices and optical computing. We also study them because they are governed by versatile mathematical models that, in addition to guided waves, describe a variety of physical systems. One may therefore understand more than one physical process by investigating a single mathematical model The new results obtained in this thesis can be broken into two main categories. First is the discovery and characteristation of new physical phenomena, in particular of several classes of vector solitary wave. Second is the application of a fascinating mathematical technique, the Hirota method, in the analysis of the integrable U(n) nonlinear Schrödinger equation. When accounting for polarisation, the simplest model for lightwave propagation is the vector nonlinear Schrödinger equation, one of the integrable soliton equations. More general models exhibit nonlinearly induced birefringence that breaks the symmetry required for integrability. We make an important generalisation of the concept of a solitary wave by permitting the polarisation components to propagate at different speeds. We thereby locate multipeaked bright soliton families in a range of media, that we call dynamic solitary waves on account of the beating between the polarisation components. All the non-fundamental forms of these waves appear to be unstable, so we transfer our attention to defocussing media and kink solitary waves. Here we find a bevy of stable bright-dark waves, and a new kind of wave, the polarisation domain wall, that only exists in the presence of nonlinear birefringence. This is a localised structure where the polarisation of the field switches state. It can be naturally extended into three dimensional models where we explore solitary waves and propagation dynamics. We also demonstrate the fundamental nature of the domain wall by linking it with the polarisation modulational instability of plane waves. Such are the new solitary waves discovered in this work. Our more mathematical results follow similar paths. Investigating the Manakov model for focussing Kerr media we write down the general 2-soliton solution, to our knowledge for the first time. We extract from this solution, as a special case, stationary states that are also special cases of dynamic solitary waves. In defocussing media we find a new type of soliton: the bright-dark soliton, for which we write down general N-soliton solutions. Our final results come from exploring the waveguide X-junctions that are formed by the collision of two solitons. In certain scenarios, these junctions have remarkable properties that can be fully characterised using the multi-solitons of the U(2) and U(3) nonlinear Schrödinger equations. In other cases we characterise these devices using an approximation scheme borrowed from linear waveguide theory.